Accretion discs efficiently transport angular momentum by a wide variety of as yet imperfectly understood mechanisms, with profound implications for the disc lifetime and planet formation. We discuss two different methods of angular momentum transport: first, generation of acoustic waves by mixing of inertial waves, and second, the generation of a self-sustaining magnetic field via the magnetorotational instability (MRI) which would be a source of dissipative turbulence. Previous local simulations of the MRI have shown that the dynamo changes character on addition of vertical stratification. We investigate numerically 3D hydrodynamic shearing waves with a conserved Hermitian form in an isothermal disc with vertical gravity, and describe the associated symplectic structure. We continue with a numerical investigation into the linear evolution of the MRI and the undular magnetic buoyancy instability in isolated flux regions and characterise the resultant quasi-linear EMFs as a function of height above the midplane. We combine this with an analytic description of the linear modes under an assumption of a poloidal-toroidal scale separation. Finally, we use RAMSES to perform full MHD simulations in a zero net flux shearing box, followed by spatial and a novel temporal averaging to reveal the essential structure of the dynamo. We find that inertial modes may be efficiently converted into acoustic modes for "bending waves", despite a fundamental ambiguity in the inertial mode structure. With our linear MRI and the undular magnetic buoyancy modes we find the localisation of the instability high in the atmosphere becomes determined by magnetic buoyancy rather than field strength for small enough azimuthal wavenumber, and that the critical Alfven speed below which the dynamo can operate increases with increasing distance from the midplane. We calculate analytically quasi-linear EMFs which predict both a vertical propagation of toroidal field and a method for creation of radial field. From our fully nonlinear calculations we find an electromotive force in phase with the toroidal field, which is itself 3π/2 out of phase with the radial (sheared) field at the midplane, and good agreement with our quasi-linear analytics. We have identified an efficient mechanism for generating acoustic waves in a disc. In our investigation of the accretion disc dynamo, we have reproduced analytically the EMFs calculated in our simulations, given arguments based on the phase of relevant quantities, several correlation integrals and the scalings suggested by our analytic work. Our analysis contributes significantly to an explanation for the dynamo in an accretion disc.

Orientador: Prof. Dr. Rodrigo Fresneda / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2016. / O modelo de Ponzano-Regge é um modelo de gravitação quântica em três dimensões. O principal objetivo deste trabalho é apresentar os fundamentos para construção desse modelo. Buscamos introduzir conceitos necessários para entendê-lo, abordando a teoria de representações de grupos de Lie compactos, tais como: redutibilidade de uma representação, representações de produto direto, e representações no espaço de funções. Tratamos especialmente do caso particular do grupo SU(2). Nesse contexto particular, apresentamos os símbolos 3j e 6j e suas propriedades. O modelo de Ponzano-Regge descreve uma geometria tridimensional discretizada, dada em termos de uma triangulação por simplexos (tetraedros, em três dimensões), de tal forma que o comprimento de cada aresta corresponde a uma representação irredutível do grupo de Lie SU(2). Estes tetraedros são descritos como símbolos 6j, cuja fórmula assintótica possibilita a passagem ao limite clássico, levando a uma expressão para a função de partição que representa uma soma sobre geometrias em três dimensões. / The Ponzano-Regge model is a quantum gravity model in three dimensions. The main goal of this work is to present the foundations for the construction of this model. We aim at introducing the necessary concepts to understand it, taking into account the theory of representations of compact Lie groups, such as: reducibility of representations, direct product representations, and representations in function spaces. We treat the particular case of the SU(2) group. In this special case, we present the 3j and 6j symbols and their properties. The Ponzano-Regge model describes a discretized 3-geometry, given in terms of a triangulation through simplices (tetrahedrons, in 3 dimensions), such that the length of each edge corresponds to an irreducible representation of the Lie group SU(2). These tetrahedrons are described as 6j symbols, whose asymptotic formula allows taking the classical limit, leading to an expression of the partition function that represents a sum over 3-geometries.